what is the slantheight of the largest square pyramid that can be curved out from a wooden cube of side 12cm?find the lateral surface area of the pyramid so obtained?
It should be close to 23m
To find the slant height of the largest square pyramid that can be curved out from a wooden cube, we need to consider the space diagonal of the cube, as it will denote the slant height of the pyramid.
The space diagonal can be found using the Pythagorean theorem:
The formula for calculating the space diagonal is √(a^2 + a^2 + a^2), where 'a' represents the length of the side of the cube.
In this case, the side length 'a' is given as 12 cm. Thus, we can calculate the space diagonal using the formula:
Space diagonal = √(12^2 + 12^2 + 12^2)
= √(144 + 144 + 144)
= √(432)
≈ 20.784 cm (rounded to three decimal places)
Therefore, the slant height of the largest square pyramid that can be curved out from the wooden cube is approximately 20.784 cm.
To find the lateral surface area of the pyramid, we need to know the slant height and the perimeter of the base.
For a square pyramid, the slant height is related to the height (h) and the side length (s) of the pyramid by the Pythagorean theorem:
Slant height^2 = height^2 + (side length/2)^2
Therefore, the height (h) can be found as:
height = √(slant height^2 - (side length/2)^2)
In this case, the side length (s) of the pyramid is equal to the side length of the base, which is 12 cm. And the slant height (l) is approximately 20.784 cm.
To find the height (h):
height = √(20.784^2 - (12/2)^2)
= √(431.712 - 36)
= √(395.712)
≈ 19.89 cm (rounded to two decimal places)
Now that we have the height and the side length of the base, we can calculate the perimeter of the base as 4 * s (since it is a square pyramid):
Perimeter of the base = 4 * 12 cm
= 48 cm
Finally, we can calculate the lateral surface area of the pyramid using the formula:
Lateral surface area = (1/2) * perimeter of base * slant height
= (1/2) * 48 cm * 20.784 cm
≈ 498.768 cm² (rounded to three decimal places)
Therefore, the lateral surface area of the pyramid obtained from the wooden cube is approximately 498.768 cm².